Rate Constant Calculation from Four Temperature and Moisture Combinations

Experimental determination of the dependance of a reaction rate constant on both temperature and moisture is laborious and time consuming by conventional methods. This Demonstration describes a method to calculate the underlying kinetic parameters from a set of four experimental determinations, and thus estimate the rate constant at any temperature-moisture combination within a desired range. The mathematical model used is an exponential temperature-dependence model, a simpler substitute for the Arrhenius equation, in which the two-parameter moisture dependence is described by a similar two-parameter exponential model. The four parameters of the resulting rate constant-temperature-moisture relationship are extracted by numerically solving four simultaneous nonlinear equations with the built-in Mathematica FindRoot function. These extracted parameters are then used to plot a 3D surface depicting the rate constant-temperature-moisture relationship and to calculate and display the rate constant for any chosen temperature-moisture combination.

THINGS TO TRY

SNAPSHOTS

DETAILS

Snapshot 1: same control settings as the Thumbnail after parameter estimation with the FindRoot function

Snapshot 2: rate constant-temperature-moisture control settings that have no FindRoot solution

Snapshot 3: hypothetical control settings for high sensitivity to temperature and to moisture

Snapshot 4: hypothetical control settings for low sensitivity to temperature and to moisture

The rate of many deteriorative reactions in stored dry foods and pharmaceuticals depends on both their temperature and moisture content. However, calculating and mapping the rate constant-temperature-moisture relationship by traditional methods requires the experimental determination of many degradation curves under constant temperature and moisture conditions, which can become a burdensome logistic issue when many different products are to be tested.

A potential shortcut method is shown to estimate the relationship from only four such curves, assuming that it can be characterized and mapped by an empirical four-parameter model in the pertinent temperature and moisture content ranges.

Assume that the relation can be described by the nested model

,

where

and are a chosen reference temperature in °C and moisture content on a dry or wet basis, respectively, and

, , and are the adjustable parameters [1].

Suppose the rate constant has been experimentally determined by a traditional method at four different sufficiently spaced constant temperature-moisture combinations yielding the values , , and . If the model indeed captures the relationship, then for the chosen and we can write four simultaneous nonlinear equations:

,

,

,

,

having four unknowns, which are the model parameters , , and .

These four simultaneous equations can be solved numerically using the FindRoot function to extract the four desired parameters. Once estimated, these parameter values together with the chosen values of and can be used to calculate and plot the relationship and to estimate the rate constant's value at any chosen temperature-moisture combination within the relevant range.

Use the sliders to set the experimental values of , , and , their corresponding temperatures and moisture content, and the selected and . These are used to calculate and display the four model parameters and plot the relationship as a 3D surface.

When a solution is reached, the slider positions for , , and indicate their calculated values, which can be used as initial guesses in subsequent calculations.

Not every possible value entered renders meaningful values of the model parameters , , and , starting from their default slider settings. When this happens, the message "No solution found!" is displayed. In such a case, you can try to adjust the kinetic parameter values by moving their sliders, guided by the effect of their settings on the 3D surface. When this fails, the conclusion is that either the model is inapplicable and/or that there is a substantial error in one or more of the entered data point coordinates.

For each parameter setting, the program also calculates and displays the rate constant's magnitude for any chosen pair of and values that are entered with the bottom two sliders. Its position is shown as a movable black dot on the 3D surface.

This Demonstration describes only the concept and mathematical methodology. The applicability of the model and methods to an actual reaction in a particular food or pharmaceutical product should be validated by testing its predictions against experimental observations not used in the kinetic parameters calculation.